understanding minimal unsatisfiability (extended abstract on …csoliver/artikel/2016_leeds.pdf ·...

Understanding minimal unsatisfiability(Extended Abstract on some recent developments)

Oliver Kullmann

Computer Science DepartmentSwansea University

Logic Colloquium 2016August 1, 2016

O Kullmann (Swansea) Understanding MU 1/8/2016 1 / 17

http://cs.swan.ac.uk/~csoliver/

http://www.swansea.ac.uk/compsci/

http://www.swansea.ac.uk/

http://www.lc2016.leeds.ac.uk/home.html

Basics

MU

A literal is a variable or a negated variable.A clause is a finite set of literals, as their disjunction; we disallowcomplementary literals here.A clause-set is a set of clauses, as their conjunction; only finiteclause-sets considered here.MU is the set of clause-sets, which are unsatisfiable, whileremoval of any clause renders them satisfiable.n(F ) is the number of (occurring) variables.c(F ) is the number of clauses.δ(F ) := c(F )− n(F ) ∈ Z is the deficiency.

See Handbook Chapter Kleine Büning and Kullmann [8].


Basics

“Tarsi’s Lemma”

For example {{a}, {a,b}, {a,b, c}, {a,b, c}

}∈MU

with δ(F ) = 4− 3 = 1.

∀F ∈MU : δ(F ) ≥ 1.

Best known proof Aharoni and Linial [1].For an overview see the introduction of [12].

Deficiency for MU yields a complexity parameter.

MU decision poly-time for fixed k (Fleischner, Kullmann, andSzeider [3]).Indeed fpt (Szeider [13]).


The main conjecture

Degrees

The most basic information about MU is givenby some knowledge on the degrees.

Literal degrees:ldF (x) := |{C ∈ F : x ∈ C}|

Variable degrees:vdF (v) := ldF (v) + ldF (v).


The main conjecture

MU(1) I

MUδ=1 = {F ∈MU : δ(F ) = 1}

These are nice formulas, with a surprising number of applications.In the SAT world, classification due to Aharoni and Linial [1],Davydov, Davydova, and Kleine Büning [2].Indeed, independently equivalent classifications in different areashave been obtained; see [12].

Complete generation process:1 Start with F = {⊥} ∈ MUδ=1.2 For F ∈MUδ=1:

1 choose C ∈ F and a variable v /∈ var(F ),2 choose D1,D2 with D1 ∪ D2 = C,3 replace C by D1 ∪ {v}, D2 ∪ {v}.


The main conjecture

MU(1) II

Characterisation becomes MUCH easier, once you know thatfor all F ∈MUδ=1, n(F ) 6= 0, there exists

v ∈ var(F ) with vdF (v) ≤ 2.

We express this asVDM(1) = 2.

Of course, there are relevant open problems already here!

For example concerning the uniform elements ofMUδ=1; Hooryand Szeider [6], Gebauer, Szabo, and Tardos [4].

Uniformity (constant clause-length) features a lot in hypergraphtheory, while we work mostly in the unrestricted setting.


The main conjecture

MU(2) I

MUδ=2 = {F ∈MU : δ(F ) = 2}

The basic characterisation is due to Kleine Büning [7].

This concerns nonsingular elements ofMUδ=2 — every variableoccurs at least twice positively as well as negatively.

{{v1 → v2}, . . . , {vn−1 → vn}, {vn → v1}

{v1, . . . , vn}, {v1, . . . , vn}}


The main conjecture

MU(2) II

The main open question here is:

Extend the generalisation to all ofMUδ=2 —in a sense fusing the characterisations obtained for δ = 1,2.

This is needed for a better understanding of higher deficiencies.

Again, a fundamental step is to to show∀F ∈MUδ=2,n(F ) 6= 0 ∃v ∈ var(F ) : vdF (v) ≤ 4.

I.e., VDM(2) = 4.


The main conjecture

Finite Patterns Conjecture

For every k ,MUδ=k can be characterised by

finitely many patterns.The next frontier isMUδ=3.


Minimum var-degree

Min-var-degree

In general (F a clause-set, C a class of clause-sets):

µvd(F ) := minv∈var(F )

vdF (v)

µvd(C) := maxF∈C

µvd(F )

VDM(k) := µvd(MUδ=k ).

In [9] the fundamental bound

∀ k ≥ 1 : VDM(k) ≤ 2k

was shown.


Minimum var-degree

Improving the bound

In [10] the upper bound VDM(k) ≤ 2k was improved to

VDM(k) ≤ 1 + k + log2(k).

Indeed a precise number-theoretical function nM(k) yields theupper bound.This upper bound is not sharp, and the first deficiency needing acorrection is k = 6.

The main open problem here is theprecise determination of VDM(k).

The sharpenings we produced unearth interesting aspects of MU; see[12] for further information.


Full clauses

Full clauses

An interesting combinatorial quantity for a clause-set F is the numberof full clauses:

fc(F ) := |{C ∈ F : var(C) = var(F )}| ∈ N0.

We havefc(F ) ≤ µvd(F ).

So maximising the number of full clauses yields lower bounds onVDM(k).

Let FCM(k) be the maximum of fc(F ) for F ∈MUδ=k .

Thus FCM ≤ VDM.


Full clauses

Hitting clause-sets

Indeed it helps a lot to consider hitting clause-sets here:

Every two clauses have a clash.

If a hitting clause-set is unsatisfiable, it is automatically MU.

VDH(k),FCH(k) denote themaximal min-var-degree resp. number of full clauses

for hitting MU.

We conjecture VDH = VDM.But definitely only FCH ≤ FCM.


Full clauses

Meta-Fibonacci

We showS2 ≤ FCH

for the number-theoretic function S2 ([11]).

And indeed we conjecture equality.

Interesting recursion-theoretic phenomena show up (at themicroscopic level).Belong to the field of “meta-Fibonacci” functions (nested recursivecalls), as introduced by Hofstadter [5].


Full clauses

The four fundamental quantities

To summarise:

The quantities VDM(k),FCM(k),VDH(k),FCH(k)seem interesting beasts: offering a lot of depth

— and good attack points!

The precise quantities matter here, and relevant number-theoreticalfunctions appear.


Conclusion

Summary and outlook

I The main conjecture is the Finite Patterns Conjecture. Once wesettled δ = 3, we hopefully see better.

II Studying the “four fundamental quantities” reveals surprisingstructures — if you go for the exact determination.

Outlook (no time to discuss in this talk):I Reductions are an important tool for understanding MU (for

example eliminating singular variables).II First you concentrate on understanding (only) reduced cases, and

then you extend.III The reductions have good properties, and likely there is much

more to come.


Conclusion

End

(references on the remaining slides).

For my papers seehttp://cs.swan.ac.uk/~csoliver/papers.html.


http://cs.swan.ac.uk/~csoliver/papers.html

Conclusion

Bibliography I

[1] Ron Aharoni and Nathan Linial. Minimal non-two-colorablehypergraphs and minimal unsatisfiable formulas. Journal ofCombinatorial Theory, Series A, 43(2):196–204, November 1986.doi:10.1016/0097-3165(86)90060-9.

[2] Gennady Davydov, Inna Davydova, and Hans Kleine Büning. Anefficient algorithm for the minimal unsatisfiability problem for asubclass of CNF. Annals of Mathematics and ArtificialIntelligence, 23(3-4):229–245, 1998.doi:10.1023/A:1018924526592.

[3] Herbert Fleischner, Oliver Kullmann, and Stefan Szeider.Polynomial–time recognition of minimal unsatisfiable formulaswith fixed clause–variable difference. Theoretical ComputerScience, 289(1):503–516, November 2002.doi:10.1016/S0304-3975(01)00337-1.


http://dx.doi.org/10.1016/0097-3165(86)90060-9

http://dx.doi.org/10.1023/A:1018924526592

http://dx.doi.org/10.1016/S0304-3975(01)00337-1

Conclusion

Bibliography II

[4] Heidi Gebauer, Tibor Szabo, and Gabor Tardos. The LocalLemma is asymptotically tight for SAT. Technical ReportarXiv:1006.0744v3 [math.CO], arXiv.org, April 2016. URLhttp://arxiv.org/abs/1006.0744.

[5] Douglas R. Hofstadter. Gödel, Escher, Bach: An eternal goldenbraid. Basic Books, 1979. ISBN 0140055797. URL http://www.physixfan.com/wp-content/files/GEBen.pdf.Pdf version with 801 pages,md5sum=“0cb32e8ea5dd2485f63842f5acffb3f0 GEBen.pdf”.

[6] Shlomo Hoory and Stefan Szeider. Computing unsatisfiablek -SAT instances with few occurrences per variable. TheoreticalComputer Science, 337(1-3):347–359, June 2005.doi:10.1016/j.tcs.2005.02.004.


http://arxiv.org/abs/1006.0744

http://www.physixfan.com/wp-content/files/GEBen.pdf

http://www.physixfan.com/wp-content/files/GEBen.pdf

http://dx.doi.org/10.1016/j.tcs.2005.02.004

Conclusion

Bibliography III

[7] Hans Kleine Büning. On subclasses of minimal unsatisfiableformulas. Discrete Applied Mathematics, 107(1-3):83–98, 2000.doi:10.1016/S0166-218X(00)00245-6.

[8] Hans Kleine Büning and Oliver Kullmann. Minimal unsatisfiabilityand autarkies. In Armin Biere, Marijn J.H. Heule, Hans vanMaaren, and Toby Walsh, editors, Handbook of Satisfiability,volume 185 of Frontiers in Artificial Intelligence and Applications,chapter 11, pages 339–401. IOS Press, February 2009. ISBN978-1-58603-929-5. doi:10.3233/978-1-58603-929-5-339.

[9] Oliver Kullmann. An application of matroid theory to the SATproblem. In Proceedings of the 15th Annual IEEE Conference onComputational Complexity, pages 116–124, July 2000.doi:10.1109/CCC.2000.856741. See also TR00-018, ElectronicColloquium on Computational Complexity (ECCC), March 2000.


http://dx.doi.org/10.1016/S0166-218X(00)00245-6

http://dx.doi.org/10.3233/978-1-58603-929-5-339

http://dx.doi.org/10.1109/CCC.2000.856741

Conclusion

Bibliography IV

[10] Oliver Kullmann and Xishun Zhao. On variables with fewoccurrences in conjunctive normal forms. In Laurent Simon andKarem Sakallah, editors, Theory and Applications of SatisfiabilityTesting - SAT 2011, volume 6695 of Lecture Notes in ComputerScience, pages 33–46. Springer, 2011. ISBN 978-3-642-14185-0.doi:10.1007/978-3-642-21581-0_5.

[11] Oliver Kullmann and Xishun Zhao. Parameters for minimalunsatisfiability: Smarandache primitive numbers and full clauses.Technical Report arXiv:1505.02318v2 [cs.DM], arXiv, July 2015.URL http://arxiv.org/abs/1505.02318.

[12] Oliver Kullmann and Xishun Zhao. Bounds for variables with fewoccurrences in conjunctive normal forms. Technical ReportarXiv:1408.0629 [math.CO], arXiv, April 2016. URLhttp://arxiv.org/abs/1408.0629.


http://dx.doi.org/10.1007/978-3-642-21581-0_5



Conclusion

Bibliography V

[13] Stefan Szeider. Minimal unsatisfiable formulas with boundedclause-variable difference are fixed-parameter tractable. Journalof Computer and System Sciences, 69(4):656–674, December2004. doi:10.1016/j.jcss.2004.04.009.


http://dx.doi.org/10.1016/j.jcss.2004.04.009

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